Since \(\rho _k\)  is a 2-core (i.e., a partition without a 2-hook),
the character \([\rho _k]\)  itself is a 2-projective irreducible character
(see [1]}).
Furthermore, any 2-regular element \(g\in S_n\)  is conjugate to its square \(g^2\) .
Hence we can apply Lemma REF  to
\(\vartheta =[\rho _k]\) , \(\chi =[\lambda ]\)  and \(p=2\) , and we obtain
the result immediately.
